We note that precisely the same distribution integral for \(\sigma ^{\tilde{T}}_{(a\mu )}\)  as in (REF ) was found utilizing the Landau level approach [1]}, there just with non-trivial spatial \(e^m_{\mu }\) , as applied to finite temperatures [2]}, [3]}. In hindsight, the same conclusion follows of course trivially from the CVE with \(e^0_i = u_i\)  and the assumption that torsion is non-zero (i.e. \(\hat{\omega }_\mu = 0\) ). This connection was first utilized in [4]}. The main result of [5]} is that

V5(=5=0) = Tr(TA)R - Tr(TA)L = dA

is the gravitational anomaly coefficient for \(N_f\)  fermions with \(T_{A}\)  the generators of a global symmetry \(G\) . For a pair of U(1)\(_{\chi }\)  Weyl fermions with opposite chirality \(\chi =\pm 1\) , we recover the axial
\(\sigma ^{\tilde{T}}_{5} = \sigma _{5}^V = \frac{\mu ^2+\mu _5^2}{4\pi ^2} + \frac{T^2}{12}\) , see below.
